The base case (n=1), given u0=5: u1=u0+21=5+2=7, but the proposed formula for u1=3+22=7. The base case is therefore true. (The corresponding series is 5, 7, 11, 19, 35, ...)
Generally, un+1=un+2n+1 (given), and un=3+2n+1 (proposed).
un+1-un=2n+1, and (proposed) un+1-un=(3+2n+2)-(3+2n+1)=2n+2-2n+1=2n+1(2-1)=2n+1.
Therefore, from the given formula and the proposed formula the difference between consecutive terms is identical, so the proposed formula is correct by induction.